#9057768

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(3,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32);(1,2,4,6)(3,5,7,8)(10,14)(11,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,18,21,25)(19,22,27,23)(20,28,29,24)(26,31,32,30);(9,12)(10,11)(13,15)(14,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262009, #72262018
maximal normal subgroup of	#72262009, #72262018
maximal quotient of	#72300493

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8964964);